Nippur was not a capital. It had no dynasty of its own and does not appear to have ruled anything, and that is precisely what made it important. It was the seat of Enlil, and Enlil handed out kingship, so a ruler who wanted his claim over Sumer to be more than a claim of arms went to Nippur.
The mechanism was concrete. Kings from Sargon of Akkad onward rebuilt and endowed the Ekur, Enlil's temple, recorded the fact in inscriptions, and installed family members in its priesthood. Sargon made his daughter Enheduanna high priestess at Ur, and the practice of placing royal daughters in the great southern temples continued for centuries. The Sumerian King List, itself a composition associated with Nippur, presents rule over the land as a single office passing from city to city, which is a theological claim as much as a historical one: at any moment one place holds kingship, and it holds it because Enlil gave it.
The arrangement gave Sumer something like a neutral centre for legitimacy, distinct from whichever city currently had the army. It also gave Enlil's priesthood real leverage, and the literature reflects the tension. The Curse of Agade tells how Naram-Sin, Sargon's grandson, took a pickaxe to the Ekur, and how in answer Enlil brought down the Gutians from the mountains and the Akkadian empire fell. Whether Naram-Sin did anything of the kind is doubtful, and the poem was probably composed generations later, but as political theology it is unambiguous: the god who gives the mandate can withdraw it, and the temple that records the giving is also the one that records the withdrawal.